Solve frac{sqrt{310^5}}{sqrt{2.710^3}}9

$$\frac{ \sqrt{ 3 { 10 }^{ 5 } } }{ \sqrt{ 2.7 { 10 }^{ 3 } } } 9$$

Evaluate

$$30\sqrt{10}\approx 94.868329805$$

Solution Steps

Calculate 10 to the power of 5 and get 100000.

$$9\times \frac{\sqrt{3\times 100000}}{\sqrt{2.7\times 10^{3}}}$$

Multiply 3 and 100000 to get 300000.

$$9\times \frac{\sqrt{300000}}{\sqrt{2.7\times 10^{3}}}$$

Factor 300000=100^{2}\times 30. Rewrite the square root of the product \sqrt{100^{2}\times 30} as the product of square roots \sqrt{100^{2}}\sqrt{30}. Take the square root of 100^{2}.

$$9\times \frac{100\sqrt{30}}{\sqrt{2.7\times 10^{3}}}$$

Calculate 10 to the power of 3 and get 1000.

$$9\times \frac{100\sqrt{30}}{\sqrt{2.7\times 1000}}$$

Multiply 2.7 and 1000 to get 2700.

$$9\times \frac{100\sqrt{30}}{\sqrt{2700}}$$

Factor 2700=30^{2}\times 3. Rewrite the square root of the product \sqrt{30^{2}\times 3} as the product of square roots \sqrt{30^{2}}\sqrt{3}. Take the square root of 30^{2}.

$$9\times \frac{100\sqrt{30}}{30\sqrt{3}}$$

Cancel out 10 in both numerator and denominator.

$$9\times \frac{10\sqrt{30}}{3\sqrt{3}}$$

Rationalize the denominator of \frac{10\sqrt{30}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$9\times \frac{10\sqrt{30}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$9\times \frac{10\sqrt{30}\sqrt{3}}{3\times 3}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$9\times \frac{10\sqrt{3}\sqrt{10}\sqrt{3}}{3\times 3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$9\times \frac{10\times 3\sqrt{10}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$9\times \frac{10\times 3\sqrt{10}}{9}$$

Multiply 10 and 3 to get 30.

$$9\times \frac{30\sqrt{10}}{9}$$

Divide 30\sqrt{10} by 9 to get \frac{10}{3}\sqrt{10}.

$$9\times \frac{10}{3}\sqrt{10}$$

Express 9\times \frac{10}{3} as a single fraction.

$$\frac{9\times 10}{3}\sqrt{10}$$

Multiply 9 and 10 to get 90.

$$\frac{90}{3}\sqrt{10}$$

Divide 90 by 3 to get 30.

$$30\sqrt{10}$$